Complex division, real part

Percentage Accurate: 70.1% → 89.6%

Time: 20.4s

Alternatives: 9

Speedup: 2.1×

• could not determine a ground truth

  1. a = -8.30496528652601e+238
  2. b = -1.677320911023856e-155
  3. c = 5285399815295833000.0
  4. d = 1.8362888823028333e+124

Specification

Math

\[\begin{array}{l} \\ \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))

C

double code(double a, double b, double c, double d) {
	return ((a * c) + (b * d)) / ((c * c) + (d * d));
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = ((a * c) + (b * d)) / ((c * c) + (d * d))
end function

Java

public static double code(double a, double b, double c, double d) {
	return ((a * c) + (b * d)) / ((c * c) + (d * d));
}

Python

def code(a, b, c, d):
	return ((a * c) + (b * d)) / ((c * c) + (d * d))

Julia

function code(a, b, c, d)
	return Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 70.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))

C

double code(double a, double b, double c, double d) {
	return ((a * c) + (b * d)) / ((c * c) + (d * d));
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = ((a * c) + (b * d)) / ((c * c) + (d * d))
end function

Java

public static double code(double a, double b, double c, double d) {
	return ((a * c) + (b * d)) / ((c * c) + (d * d));
}

Python

def code(a, b, c, d):
	return ((a * c) + (b * d)) / ((c * c) + (d * d))

Julia

function code(a, b, c, d)
	return Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 89.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.55 \cdot 10^{+27}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{\frac{d \cdot d}{a}}\\ \mathbf{elif}\;d \leq -8 \cdot 10^{-143}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b \cdot d}{c}\right)\\ \mathbf{elif}\;d \leq 19500:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{a \cdot c}{d \cdot d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.55e+27)
   (+ (/ b d) (/ c (/ (* d d) a)))
   (if (<= d -8e-143)
     (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
     (if (<= d 4.5e-115)
       (* (/ 1.0 c) (+ a (/ (* b d) c)))
       (if (<= d 19500.0)
         (/ (fma a c (* b d)) (fma c c (* d d)))
         (+ (/ b d) (/ (* a c) (* d d))))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.55e+27) {
		tmp = (b / d) + (c / ((d * d) / a));
	} else if (d <= -8e-143) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (d <= 4.5e-115) {
		tmp = (1.0 / c) * (a + ((b * d) / c));
	} else if (d <= 19500.0) {
		tmp = fma(a, c, (b * d)) / fma(c, c, (d * d));
	} else {
		tmp = (b / d) + ((a * c) / (d * d));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.55e+27)
		tmp = Float64(Float64(b / d) + Float64(c / Float64(Float64(d * d) / a)));
	elseif (d <= -8e-143)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 4.5e-115)
		tmp = Float64(Float64(1.0 / c) * Float64(a + Float64(Float64(b * d) / c)));
	elseif (d <= 19500.0)
		tmp = Float64(fma(a, c, Float64(b * d)) / fma(c, c, Float64(d * d)));
	else
		tmp = Float64(Float64(b / d) + Float64(Float64(a * c) / Float64(d * d)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.55e+27], N[(N[(b / d), $MachinePrecision] + N[(c / N[(N[(d * d), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -8e-143], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.5e-115], N[(N[(1.0 / c), $MachinePrecision] * N[(a + N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 19500.0], N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b / d), $MachinePrecision] + N[(N[(a * c), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.54999999999999998e27

   1. Initial program 72.7%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around 0 97.1%

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{c \cdot a}{{d}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 97.1%

         \[\leadsto \frac{b}{d} + \frac{c \cdot a}{\color{blue}{d \cdot d}} \]

      2. associate-/l* 97.1%

         \[\leadsto \frac{b}{d} + \color{blue}{\frac{c}{\frac{d \cdot d}{a}}} \]

   4. Simplified 97.1%

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{c}{\frac{d \cdot d}{a}}} \]

   if -1.54999999999999998e27 < d < -7.9999999999999996e-143

   1. Initial program 87.4%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   if -7.9999999999999996e-143 < d < 4.50000000000000023e-115

   1. Initial program 70.1%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Step-by-step derivation

      1. *-un-lft-identity 70.1%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{c \cdot c + d \cdot d} \]

      2. add-sqr-sqrt 70.1%

         \[\leadsto \frac{1 \cdot \left(a \cdot c + b \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]

      3. times-frac 70.2%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]

      4. hypot-def 70.2%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]

      5. fma-def 70.2%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d}} \]

      6. hypot-def 85.5%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

   3. Applied egg-rr 85.5%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}} \]

   4. Taylor expanded in c around inf 49.0%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(a + \frac{d \cdot b}{c}\right)} \]

   5. Taylor expanded in c around inf 91.7%

      \[\leadsto \color{blue}{\frac{1}{c}} \cdot \left(a + \frac{d \cdot b}{c}\right) \]

   if 4.50000000000000023e-115 < d < 19500

   1. Initial program 87.8%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Step-by-step derivation

      1. fma-def 87.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right)}}{c \cdot c + d \cdot d} \]

      2. fma-def 87.9%

         \[\leadsto \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\mathsf{fma}\left(c, c, d \cdot d\right)}} \]

   3. Simplified 87.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}} \]

   if 19500 < d

   1. Initial program 73.6%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around 0 98.3%

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{c \cdot a}{{d}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 98.3%

         \[\leadsto \frac{b}{d} + \frac{c \cdot a}{\color{blue}{d \cdot d}} \]

      2. times-frac 98.2%

         \[\leadsto \frac{b}{d} + \color{blue}{\frac{c}{d} \cdot \frac{a}{d}} \]

   4. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}} \]
   5. Step-by-step derivation

      1. frac-times 98.3%

         \[\leadsto \frac{b}{d} + \color{blue}{\frac{c \cdot a}{d \cdot d}} \]

   6. Applied egg-rr 98.3%

      \[\leadsto \frac{b}{d} + \color{blue}{\frac{c \cdot a}{d \cdot d}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.55 \cdot 10^{+27}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{\frac{d \cdot d}{a}}\\ \mathbf{elif}\;d \leq -8 \cdot 10^{-143}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b \cdot d}{c}\right)\\ \mathbf{elif}\;d \leq 19500:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{a \cdot c}{d \cdot d}\\ \end{array} \]

Alternative 2: 88.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq \infty:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \frac{c}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) INFINITY)
   (* (/ 1.0 (hypot c d)) (/ (fma a c (* b d)) (hypot c d)))
   (/ (* a (/ c (hypot c d))) (hypot c d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((((a * c) + (b * d)) / ((c * c) + (d * d))) <= ((double) INFINITY)) {
		tmp = (1.0 / hypot(c, d)) * (fma(a, c, (b * d)) / hypot(c, d));
	} else {
		tmp = (a * (c / hypot(c, d))) / hypot(c, d);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d))) <= Inf)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(fma(a, c, Float64(b * d)) / hypot(c, d)));
	else
		tmp = Float64(Float64(a * Float64(c / hypot(c, d))) / hypot(c, d));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d))) < +inf.0

   1. Initial program 84.5%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Step-by-step derivation

      1. *-un-lft-identity 84.5%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{c \cdot c + d \cdot d} \]

      2. add-sqr-sqrt 84.5%

         \[\leadsto \frac{1 \cdot \left(a \cdot c + b \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]

      3. times-frac 84.5%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]

      4. hypot-def 84.5%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]

      5. fma-def 84.5%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d}} \]

      6. hypot-def 95.0%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

   3. Applied egg-rr 95.0%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}} \]

   if +inf.0 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 0.0%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in a around inf 1.8%

      \[\leadsto \color{blue}{\frac{c \cdot a}{{d}^{2} + {c}^{2}}} \]
   3. Step-by-step derivation

      1. associate-/l* 3.6%

         \[\leadsto \color{blue}{\frac{c}{\frac{{d}^{2} + {c}^{2}}{a}}} \]

      2. associate-/r/ 3.6%

         \[\leadsto \color{blue}{\frac{c}{{d}^{2} + {c}^{2}} \cdot a} \]

      3. unpow2 3.6%

         \[\leadsto \frac{c}{\color{blue}{d \cdot d} + {c}^{2}} \cdot a \]

      4. unpow2 3.6%

         \[\leadsto \frac{c}{d \cdot d + \color{blue}{c \cdot c}} \cdot a \]

      5. +-commutative 3.6%

         \[\leadsto \frac{c}{\color{blue}{c \cdot c + d \cdot d}} \cdot a \]

      6. fma-def 3.6%

         \[\leadsto \frac{c}{\color{blue}{\mathsf{fma}\left(c, c, d \cdot d\right)}} \cdot a \]

   4. Simplified 3.6%

      \[\leadsto \color{blue}{\frac{c}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot a} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 3.6%

         \[\leadsto \frac{\color{blue}{1 \cdot c}}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot a \]

      2. fma-def 3.6%

         \[\leadsto \frac{1 \cdot c}{\color{blue}{c \cdot c + d \cdot d}} \cdot a \]

      3. add-sqr-sqrt 3.6%

         \[\leadsto \frac{1 \cdot c}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \cdot a \]

      4. hypot-udef 3.6%

         \[\leadsto \frac{1 \cdot c}{\color{blue}{\mathsf{hypot}\left(c, d\right)} \cdot \sqrt{c \cdot c + d \cdot d}} \cdot a \]

      5. hypot-udef 3.6%

         \[\leadsto \frac{1 \cdot c}{\mathsf{hypot}\left(c, d\right) \cdot \color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot a \]

      6. times-frac 53.5%

         \[\leadsto \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c}{\mathsf{hypot}\left(c, d\right)}\right)} \cdot a \]

   6. Applied egg-rr 53.5%

      \[\leadsto \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c}{\mathsf{hypot}\left(c, d\right)}\right)} \cdot a \]
   7. Step-by-step derivation

      1. associate-*l/ 53.4%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{c}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}} \cdot a \]

      2. *-lft-identity 53.4%

         \[\leadsto \frac{\color{blue}{\frac{c}{\mathsf{hypot}\left(c, d\right)}}}{\mathsf{hypot}\left(c, d\right)} \cdot a \]

   8. Simplified 53.4%

      \[\leadsto \color{blue}{\frac{\frac{c}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}} \cdot a \]
   9. Step-by-step derivation

      1. associate-*l/ 53.7%

         \[\leadsto \color{blue}{\frac{\frac{c}{\mathsf{hypot}\left(c, d\right)} \cdot a}{\mathsf{hypot}\left(c, d\right)}} \]

   10. Applied egg-rr 53.7%

       \[\leadsto \color{blue}{\frac{\frac{c}{\mathsf{hypot}\left(c, d\right)} \cdot a}{\mathsf{hypot}\left(c, d\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq \infty:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \frac{c}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 3: 89.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -1.55 \cdot 10^{+27}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{\frac{d \cdot d}{a}}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-142}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{-115}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b \cdot d}{c}\right)\\ \mathbf{elif}\;d \leq 21000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{a \cdot c}{d \cdot d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= d -1.55e+27)
     (+ (/ b d) (/ c (/ (* d d) a)))
     (if (<= d -1e-142)
       t_0
       (if (<= d 6.8e-115)
         (* (/ 1.0 c) (+ a (/ (* b d) c)))
         (if (<= d 21000.0) t_0 (+ (/ b d) (/ (* a c) (* d d)))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -1.55e+27) {
		tmp = (b / d) + (c / ((d * d) / a));
	} else if (d <= -1e-142) {
		tmp = t_0;
	} else if (d <= 6.8e-115) {
		tmp = (1.0 / c) * (a + ((b * d) / c));
	} else if (d <= 21000.0) {
		tmp = t_0;
	} else {
		tmp = (b / d) + ((a * c) / (d * d));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
    if (d <= (-1.55d+27)) then
        tmp = (b / d) + (c / ((d * d) / a))
    else if (d <= (-1d-142)) then
        tmp = t_0
    else if (d <= 6.8d-115) then
        tmp = (1.0d0 / c) * (a + ((b * d) / c))
    else if (d <= 21000.0d0) then
        tmp = t_0
    else
        tmp = (b / d) + ((a * c) / (d * d))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -1.55e+27) {
		tmp = (b / d) + (c / ((d * d) / a));
	} else if (d <= -1e-142) {
		tmp = t_0;
	} else if (d <= 6.8e-115) {
		tmp = (1.0 / c) * (a + ((b * d) / c));
	} else if (d <= 21000.0) {
		tmp = t_0;
	} else {
		tmp = (b / d) + ((a * c) / (d * d));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if d <= -1.55e+27:
		tmp = (b / d) + (c / ((d * d) / a))
	elif d <= -1e-142:
		tmp = t_0
	elif d <= 6.8e-115:
		tmp = (1.0 / c) * (a + ((b * d) / c))
	elif d <= 21000.0:
		tmp = t_0
	else:
		tmp = (b / d) + ((a * c) / (d * d))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -1.55e+27)
		tmp = Float64(Float64(b / d) + Float64(c / Float64(Float64(d * d) / a)));
	elseif (d <= -1e-142)
		tmp = t_0;
	elseif (d <= 6.8e-115)
		tmp = Float64(Float64(1.0 / c) * Float64(a + Float64(Float64(b * d) / c)));
	elseif (d <= 21000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(b / d) + Float64(Float64(a * c) / Float64(d * d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (d <= -1.55e+27)
		tmp = (b / d) + (c / ((d * d) / a));
	elseif (d <= -1e-142)
		tmp = t_0;
	elseif (d <= 6.8e-115)
		tmp = (1.0 / c) * (a + ((b * d) / c));
	elseif (d <= 21000.0)
		tmp = t_0;
	else
		tmp = (b / d) + ((a * c) / (d * d));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.55e+27], N[(N[(b / d), $MachinePrecision] + N[(c / N[(N[(d * d), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-142], t$95$0, If[LessEqual[d, 6.8e-115], N[(N[(1.0 / c), $MachinePrecision] * N[(a + N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 21000.0], t$95$0, N[(N[(b / d), $MachinePrecision] + N[(N[(a * c), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.54999999999999998e27

   1. Initial program 72.7%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around 0 97.1%

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{c \cdot a}{{d}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 97.1%

         \[\leadsto \frac{b}{d} + \frac{c \cdot a}{\color{blue}{d \cdot d}} \]

      2. associate-/l* 97.1%

         \[\leadsto \frac{b}{d} + \color{blue}{\frac{c}{\frac{d \cdot d}{a}}} \]

   4. Simplified 97.1%

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{c}{\frac{d \cdot d}{a}}} \]

   if -1.54999999999999998e27 < d < -1e-142 or 6.7999999999999996e-115 < d < 21000

   1. Initial program 87.5%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   if -1e-142 < d < 6.7999999999999996e-115

   1. Initial program 70.1%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Step-by-step derivation

      1. *-un-lft-identity 70.1%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{c \cdot c + d \cdot d} \]

      2. add-sqr-sqrt 70.1%

         \[\leadsto \frac{1 \cdot \left(a \cdot c + b \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]

      3. times-frac 70.2%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]

      4. hypot-def 70.2%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]

      5. fma-def 70.2%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d}} \]

      6. hypot-def 85.5%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

   3. Applied egg-rr 85.5%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}} \]

   4. Taylor expanded in c around inf 49.0%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(a + \frac{d \cdot b}{c}\right)} \]

   5. Taylor expanded in c around inf 91.7%

      \[\leadsto \color{blue}{\frac{1}{c}} \cdot \left(a + \frac{d \cdot b}{c}\right) \]

   if 21000 < d

   1. Initial program 73.6%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around 0 98.3%

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{c \cdot a}{{d}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 98.3%

         \[\leadsto \frac{b}{d} + \frac{c \cdot a}{\color{blue}{d \cdot d}} \]

      2. times-frac 98.2%

         \[\leadsto \frac{b}{d} + \color{blue}{\frac{c}{d} \cdot \frac{a}{d}} \]

   4. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}} \]
   5. Step-by-step derivation

      1. frac-times 98.3%

         \[\leadsto \frac{b}{d} + \color{blue}{\frac{c \cdot a}{d \cdot d}} \]

   6. Applied egg-rr 98.3%

      \[\leadsto \frac{b}{d} + \color{blue}{\frac{c \cdot a}{d \cdot d}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.55 \cdot 10^{+27}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{\frac{d \cdot d}{a}}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-142}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{-115}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b \cdot d}{c}\right)\\ \mathbf{elif}\;d \leq 21000:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{a \cdot c}{d \cdot d}\\ \end{array} \]

Alternative 4: 83.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.8 \cdot 10^{-8} \lor \neg \left(c \leq 1.3\right):\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -4.8e-8) (not (<= c 1.3)))
   (+ (/ a c) (* (/ d c) (/ b c)))
   (+ (/ b d) (* (/ c d) (/ a d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4.8e-8) || !(c <= 1.3)) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else {
		tmp = (b / d) + ((c / d) * (a / d));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-4.8d-8)) .or. (.not. (c <= 1.3d0))) then
        tmp = (a / c) + ((d / c) * (b / c))
    else
        tmp = (b / d) + ((c / d) * (a / d))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4.8e-8) || !(c <= 1.3)) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else {
		tmp = (b / d) + ((c / d) * (a / d));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -4.8e-8) or not (c <= 1.3):
		tmp = (a / c) + ((d / c) * (b / c))
	else:
		tmp = (b / d) + ((c / d) * (a / d))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -4.8e-8) || !(c <= 1.3))
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	else
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -4.8e-8) || ~((c <= 1.3)))
		tmp = (a / c) + ((d / c) * (b / c));
	else
		tmp = (b / d) + ((c / d) * (a / d));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -4.8e-8], N[Not[LessEqual[c, 1.3]], $MachinePrecision]], N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -4.79999999999999997e-8 or 1.30000000000000004 < c

   1. Initial program 68.9%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around inf 94.6%

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{d \cdot b}{{c}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 94.6%

         \[\leadsto \frac{a}{c} + \frac{d \cdot b}{\color{blue}{c \cdot c}} \]

      2. times-frac 95.7%

         \[\leadsto \frac{a}{c} + \color{blue}{\frac{d}{c} \cdot \frac{b}{c}} \]

   4. Simplified 95.7%

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}} \]

   if -4.79999999999999997e-8 < c < 1.30000000000000004

   1. Initial program 79.0%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around 0 75.7%

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{c \cdot a}{{d}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 75.7%

         \[\leadsto \frac{b}{d} + \frac{c \cdot a}{\color{blue}{d \cdot d}} \]

      2. times-frac 80.3%

         \[\leadsto \frac{b}{d} + \color{blue}{\frac{c}{d} \cdot \frac{a}{d}} \]

   4. Simplified 80.3%

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.8 \cdot 10^{-8} \lor \neg \left(c \leq 1.3\right):\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \end{array} \]

Alternative 5: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.3 \cdot 10^{-8} \lor \neg \left(c \leq 0.000235\right):\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d \cdot \frac{d}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -5.3e-8) (not (<= c 0.000235)))
   (+ (/ a c) (* (/ d c) (/ b c)))
   (+ (/ b d) (/ a (* d (/ d c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -5.3e-8) || !(c <= 0.000235)) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else {
		tmp = (b / d) + (a / (d * (d / c)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-5.3d-8)) .or. (.not. (c <= 0.000235d0))) then
        tmp = (a / c) + ((d / c) * (b / c))
    else
        tmp = (b / d) + (a / (d * (d / c)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -5.3e-8) || !(c <= 0.000235)) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else {
		tmp = (b / d) + (a / (d * (d / c)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -5.3e-8) or not (c <= 0.000235):
		tmp = (a / c) + ((d / c) * (b / c))
	else:
		tmp = (b / d) + (a / (d * (d / c)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -5.3e-8) || !(c <= 0.000235))
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	else
		tmp = Float64(Float64(b / d) + Float64(a / Float64(d * Float64(d / c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -5.3e-8) || ~((c <= 0.000235)))
		tmp = (a / c) + ((d / c) * (b / c));
	else
		tmp = (b / d) + (a / (d * (d / c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -5.3e-8], N[Not[LessEqual[c, 0.000235]], $MachinePrecision]], N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b / d), $MachinePrecision] + N[(a / N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -5.2999999999999998e-8 or 2.34999999999999993e-4 < c

   1. Initial program 68.9%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around inf 94.6%

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{d \cdot b}{{c}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 94.6%

         \[\leadsto \frac{a}{c} + \frac{d \cdot b}{\color{blue}{c \cdot c}} \]

      2. times-frac 95.7%

         \[\leadsto \frac{a}{c} + \color{blue}{\frac{d}{c} \cdot \frac{b}{c}} \]

   4. Simplified 95.7%

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}} \]

   if -5.2999999999999998e-8 < c < 2.34999999999999993e-4

   1. Initial program 79.0%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around 0 75.7%

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{c \cdot a}{{d}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 75.7%

         \[\leadsto \frac{b}{d} + \frac{c \cdot a}{\color{blue}{d \cdot d}} \]

      2. times-frac 80.3%

         \[\leadsto \frac{b}{d} + \color{blue}{\frac{c}{d} \cdot \frac{a}{d}} \]

   4. Simplified 80.3%

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}} \]
   5. Step-by-step derivation

      1. clear-num 79.8%

         \[\leadsto \frac{b}{d} + \color{blue}{\frac{1}{\frac{d}{c}}} \cdot \frac{a}{d} \]

      2. frac-times 80.3%

         \[\leadsto \frac{b}{d} + \color{blue}{\frac{1 \cdot a}{\frac{d}{c} \cdot d}} \]

      3. *-un-lft-identity 80.3%

         \[\leadsto \frac{b}{d} + \frac{\color{blue}{a}}{\frac{d}{c} \cdot d} \]

   6. Applied egg-rr 80.3%

      \[\leadsto \frac{b}{d} + \color{blue}{\frac{a}{\frac{d}{c} \cdot d}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.3 \cdot 10^{-8} \lor \neg \left(c \leq 0.000235\right):\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d \cdot \frac{d}{c}}\\ \end{array} \]

Alternative 6: 82.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -8200000000000:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 70000000:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b \cdot d}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -8200000000000.0)
   (/ b d)
   (if (<= d 70000000.0) (* (/ 1.0 c) (+ a (/ (* b d) c))) (/ b d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -8200000000000.0) {
		tmp = b / d;
	} else if (d <= 70000000.0) {
		tmp = (1.0 / c) * (a + ((b * d) / c));
	} else {
		tmp = b / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-8200000000000.0d0)) then
        tmp = b / d
    else if (d <= 70000000.0d0) then
        tmp = (1.0d0 / c) * (a + ((b * d) / c))
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -8200000000000.0) {
		tmp = b / d;
	} else if (d <= 70000000.0) {
		tmp = (1.0 / c) * (a + ((b * d) / c));
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -8200000000000.0:
		tmp = b / d
	elif d <= 70000000.0:
		tmp = (1.0 / c) * (a + ((b * d) / c))
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -8200000000000.0)
		tmp = Float64(b / d);
	elseif (d <= 70000000.0)
		tmp = Float64(Float64(1.0 / c) * Float64(a + Float64(Float64(b * d) / c)));
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -8200000000000.0)
		tmp = b / d;
	elseif (d <= 70000000.0)
		tmp = (1.0 / c) * (a + ((b * d) / c));
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -8200000000000.0], N[(b / d), $MachinePrecision], If[LessEqual[d, 70000000.0], N[(N[(1.0 / c), $MachinePrecision] * N[(a + N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -8.2e12 or 7e7 < d

   1. Initial program 73.0%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around 0 87.0%

      \[\leadsto \color{blue}{\frac{b}{d}} \]

   if -8.2e12 < d < 7e7

   1. Initial program 76.9%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Step-by-step derivation

      1. *-un-lft-identity 76.9%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{c \cdot c + d \cdot d} \]

      2. add-sqr-sqrt 76.9%

         \[\leadsto \frac{1 \cdot \left(a \cdot c + b \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]

      3. times-frac 76.9%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]

      4. hypot-def 76.9%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]

      5. fma-def 76.9%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d}} \]

      6. hypot-def 88.1%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

   3. Applied egg-rr 88.1%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}} \]

   4. Taylor expanded in c around inf 44.7%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(a + \frac{d \cdot b}{c}\right)} \]

   5. Taylor expanded in c around inf 79.1%

      \[\leadsto \color{blue}{\frac{1}{c}} \cdot \left(a + \frac{d \cdot b}{c}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8200000000000:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 70000000:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b \cdot d}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 7: 84.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -45:\\ \;\;\;\;\frac{b}{d} + \frac{c}{\frac{d \cdot d}{a}}\\ \mathbf{elif}\;d \leq 1.12 \cdot 10^{-79}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b \cdot d}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{a \cdot c}{d \cdot d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -45.0)
   (+ (/ b d) (/ c (/ (* d d) a)))
   (if (<= d 1.12e-79)
     (* (/ 1.0 c) (+ a (/ (* b d) c)))
     (+ (/ b d) (/ (* a c) (* d d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -45.0) {
		tmp = (b / d) + (c / ((d * d) / a));
	} else if (d <= 1.12e-79) {
		tmp = (1.0 / c) * (a + ((b * d) / c));
	} else {
		tmp = (b / d) + ((a * c) / (d * d));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-45.0d0)) then
        tmp = (b / d) + (c / ((d * d) / a))
    else if (d <= 1.12d-79) then
        tmp = (1.0d0 / c) * (a + ((b * d) / c))
    else
        tmp = (b / d) + ((a * c) / (d * d))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -45.0) {
		tmp = (b / d) + (c / ((d * d) / a));
	} else if (d <= 1.12e-79) {
		tmp = (1.0 / c) * (a + ((b * d) / c));
	} else {
		tmp = (b / d) + ((a * c) / (d * d));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -45.0:
		tmp = (b / d) + (c / ((d * d) / a))
	elif d <= 1.12e-79:
		tmp = (1.0 / c) * (a + ((b * d) / c))
	else:
		tmp = (b / d) + ((a * c) / (d * d))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -45.0)
		tmp = Float64(Float64(b / d) + Float64(c / Float64(Float64(d * d) / a)));
	elseif (d <= 1.12e-79)
		tmp = Float64(Float64(1.0 / c) * Float64(a + Float64(Float64(b * d) / c)));
	else
		tmp = Float64(Float64(b / d) + Float64(Float64(a * c) / Float64(d * d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -45.0)
		tmp = (b / d) + (c / ((d * d) / a));
	elseif (d <= 1.12e-79)
		tmp = (1.0 / c) * (a + ((b * d) / c));
	else
		tmp = (b / d) + ((a * c) / (d * d));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -45.0], N[(N[(b / d), $MachinePrecision] + N[(c / N[(N[(d * d), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.12e-79], N[(N[(1.0 / c), $MachinePrecision] * N[(a + N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b / d), $MachinePrecision] + N[(N[(a * c), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -45

   1. Initial program 74.5%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around 0 90.3%

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{c \cdot a}{{d}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 90.3%

         \[\leadsto \frac{b}{d} + \frac{c \cdot a}{\color{blue}{d \cdot d}} \]

      2. associate-/l* 90.3%

         \[\leadsto \frac{b}{d} + \color{blue}{\frac{c}{\frac{d \cdot d}{a}}} \]

   4. Simplified 90.3%

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{c}{\frac{d \cdot d}{a}}} \]

   if -45 < d < 1.11999999999999996e-79

   1. Initial program 75.2%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Step-by-step derivation

      1. *-un-lft-identity 75.2%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{c \cdot c + d \cdot d} \]

      2. add-sqr-sqrt 75.1%

         \[\leadsto \frac{1 \cdot \left(a \cdot c + b \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]

      3. times-frac 75.2%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]

      4. hypot-def 75.2%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]

      5. fma-def 75.2%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d}} \]

      6. hypot-def 86.9%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

   3. Applied egg-rr 86.9%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}} \]

   4. Taylor expanded in c around inf 46.5%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(a + \frac{d \cdot b}{c}\right)} \]

   5. Taylor expanded in c around inf 84.8%

      \[\leadsto \color{blue}{\frac{1}{c}} \cdot \left(a + \frac{d \cdot b}{c}\right) \]

   if 1.11999999999999996e-79 < d

   1. Initial program 77.0%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around 0 88.0%

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{c \cdot a}{{d}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 88.0%

         \[\leadsto \frac{b}{d} + \frac{c \cdot a}{\color{blue}{d \cdot d}} \]

      2. times-frac 86.7%

         \[\leadsto \frac{b}{d} + \color{blue}{\frac{c}{d} \cdot \frac{a}{d}} \]

   4. Simplified 86.7%

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}} \]
   5. Step-by-step derivation

      1. frac-times 88.0%

         \[\leadsto \frac{b}{d} + \color{blue}{\frac{c \cdot a}{d \cdot d}} \]

   6. Applied egg-rr 88.0%

      \[\leadsto \frac{b}{d} + \color{blue}{\frac{c \cdot a}{d \cdot d}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -45:\\ \;\;\;\;\frac{b}{d} + \frac{c}{\frac{d \cdot d}{a}}\\ \mathbf{elif}\;d \leq 1.12 \cdot 10^{-79}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b \cdot d}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{a \cdot c}{d \cdot d}\\ \end{array} \]

Alternative 8: 70.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.3 \cdot 10^{-30}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 12.6:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.3e-30) (/ a c) (if (<= c 12.6) (/ b d) (/ a c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.3e-30) {
		tmp = a / c;
	} else if (c <= 12.6) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-2.3d-30)) then
        tmp = a / c
    else if (c <= 12.6d0) then
        tmp = b / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.3e-30) {
		tmp = a / c;
	} else if (c <= 12.6) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -2.3e-30:
		tmp = a / c
	elif c <= 12.6:
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.3e-30)
		tmp = Float64(a / c);
	elseif (c <= 12.6)
		tmp = Float64(b / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -2.3e-30)
		tmp = a / c;
	elseif (c <= 12.6)
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2.3e-30], N[(a / c), $MachinePrecision], If[LessEqual[c, 12.6], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -2.29999999999999984e-30 or 12.5999999999999996 < c

   1. Initial program 70.9%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around inf 79.5%

      \[\leadsto \color{blue}{\frac{a}{c}} \]

   if -2.29999999999999984e-30 < c < 12.5999999999999996

   1. Initial program 78.2%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around 0 63.3%

      \[\leadsto \color{blue}{\frac{b}{d}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.3 \cdot 10^{-30}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 12.6:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 9: 43.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{a}{c} \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (/ a c))

C

double code(double a, double b, double c, double d) {
	return a / c;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = a / c
end function

Java

public static double code(double a, double b, double c, double d) {
	return a / c;
}

Python

def code(a, b, c, d):
	return a / c

Julia

function code(a, b, c, d)
	return Float64(a / c)
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = a / c;
end

Wolfram

code[a_, b_, c_, d_] := N[(a / c), $MachinePrecision]

Derivation

1. Initial program 75.6%

   \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

2. Taylor expanded in c around inf 40.6%

   \[\leadsto \color{blue}{\frac{a}{c}} \]

3. Final simplification 40.6%

   \[\leadsto \frac{a}{c} \]

Developer target: 99.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|d\right| < \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (< (fabs d) (fabs c))
   (/ (+ a (* b (/ d c))) (+ c (* d (/ d c))))
   (/ (+ b (* a (/ c d))) (+ d (* c (/ c d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (fabs(d) < fabs(c)) {
		tmp = (a + (b * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (abs(d) < abs(c)) then
        tmp = (a + (b * (d / c))) / (c + (d * (d / c)))
    else
        tmp = (b + (a * (c / d))) / (d + (c * (c / d)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (Math.abs(d) < Math.abs(c)) {
		tmp = (a + (b * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if math.fabs(d) < math.fabs(c):
		tmp = (a + (b * (d / c))) / (c + (d * (d / c)))
	else:
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (abs(d) < abs(c))
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / Float64(c + Float64(d * Float64(d / c))));
	else
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / Float64(d + Float64(c * Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (abs(d) < abs(c))
		tmp = (a + (b * (d / c))) / (c + (d * (d / c)));
	else
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Less[N[Abs[d], $MachinePrecision], N[Abs[c], $MachinePrecision]], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c + N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d + N[(c * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2023278 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (+ a (* b (/ d c))) (+ c (* d (/ d c)))) (/ (+ b (* a (/ c d))) (+ d (* c (/ c d)))))

  (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))